Re: optimal set of basis vectors

"jindra" <jparus@xxxxxxxxx> wrote in message
news:1174403228.384115.200470@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

To describe some real application. Let's have a set of 3D vectors V =
{vi} in E3 (analogy of aforementioned L where N = 3). Project vi using
orthogonal projection on some plane (the plane is a subspace of E3,
i.e., M = 2) so that the projection is closest possible with respect
to V. For example, we can project vectors vi to xy plane which will
yield some error, or we can project vectors vi to xz plane which will
yield another error. I would like to find such plane which would yeild
the smallest error. The question is how to find such plane, i.e., its
basis vectors. Of course I'm looking for some dimension indepedent
solution, this was just one instance of my problem.

You may use a least-squares error approach. This document,
http://www.geometrictools.com/Documentation/LeastSquaresFitting.pdf
shows how to fit a line to points in N dimensions and how to fit a
hyperplane (M = N-1) in N dimensions. The ideas easily extend
to fitting an M-flat to points in N dimensions.

--
Dave Eberly
http://www.geometrictools.com


.

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